What Is Distribution Mean?
The mean of a probability distribution is its expected value: the long-run average of the random variable. For a table of values and probabilities, you multiply each value by its probability and add.
Choose a discrete table, or pick binomial, Poisson, geometric, continuous uniform, exponential or normal and enter its parameters. The calculator returns the mean, variance and standard deviation and shows the working.
The mean can be a number that never occurs in one trial, such as 3.5 for a fair die. It is the balance point of the distribution, not a prediction for a single outcome.
How to Use the Distribution Mean Calculator
- Choose a discrete table or a standard distribution.
- Enter the values and probabilities, or the parameters.
- Read the mean, variance and standard deviation.
Distribution Mean Formula
Discrete mean
Variance
Binomial
Distribution Mean Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| μ or E[X] | Mean, the expected value | |
| σ² | Variance of the distribution | |
| P(x) | Probability of the value x | The table must add to 1 |
| n, p | Binomial parameters | μ = np |
| λ | Poisson mean or exponential rate |
Distribution Mean Worked Example
With the example values loaded in the calculator above, the steps are:
- Distribution: Discrete distribution.
- Mean: E[X] = Σ x·P(x) = 1.7.
- Variance: E[X²] − μ² = 3.7 − 2.89 = 0.81.
- Standard deviation = √0.81 = 0.9.
Result: Mean (expected value) μ = 1.7; Variance σ² = 0.81.
Worked example: a discrete table
For x = 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2 the mean is 0 × 0.1 + 1 × 0.3 + 2 × 0.4 + 3 × 0.2 = 1.7.
E[X²] = 0 + 0.3 + 1.6 + 1.8 = 3.7, so the variance is 3.7 − 1.7² = 0.81 and the standard deviation is 0.9.
What the mean of a distribution means
It is the balance point of the distribution. It is also the average you would get after a very large number of trials, which is why a mean of 1.7 is fine even though 1.7 can never occur on one trial.
Where Distribution Mean Is Used
Probability coursework
Compute E[X] and Var(X) from a table and check the working.
Games and decisions
Find the expected payoff of a game from its outcome table.
Modelling counts and waiting times
Get the mean of a binomial, Poisson, geometric or exponential model.
Common Distribution Mean Mistakes to Avoid
- Probabilities in a table must add to 1.
- Use the variance to see how spread out the outcomes are around the mean.
Distribution Mean Terms and Definitions
- Expected value
- The probability-weighted average of all possible values of a random variable.
Distribution Mean FAQs
How do you find the mean of a probability distribution?
Multiply each value by its probability and add the results.
Is the mean the same as the expected value?
Yes. For a random variable, mean and expected value are the same thing.
What is the mean of a binomial distribution?
n × p, the number of trials times the success probability.
What is the mean of a Poisson distribution?
λ, which also equals its variance.
Can the mean be a value that never occurs?
Yes. A fair die has a mean of 3.5 even though 3.5 is never rolled.
What must the probabilities add up to?
1. The calculator checks this and reports an error if they do not.